Second-sound acoustic turbulence in superfluid helium: Decay of the direct and inverse energy cascades

We have investigated the decay of second-sound acoustic turbulence in superfluid ${}^4$He following removal of the driving force that created it. A periodically driven cylindrical resonator of high quality $Q$ factor was used to create one-dimensional second sound of large wave amplitude. The resultant acoustic turbulence involved an energy cascade towards the higher frequencies (smaller scales) where viscous dissipation occurred. Under some conditions, there was also a co-existing inverse energy cascade in the opposite direction, towards lower frequencies, where dissipation also occurred. We have found that the spectral dynamics that occurs when the periodic drive is switched off exhibits complex and interesting features. We show that the wave decay starts from the high-frequency end of the spectrum in both cases, and we have been able to identify and measure a nonlinear decay time related to the interwave interactions, as distinct from the linear decay process due to dissipation. A windowed Fourier analysis reveals that the occurrence of seemingly chaotic oscillations of the individual harmonic amplitudes were observed during the decay.

Figure 1

Dependence on frequency $f$ of the resonator quality $Q$ factor at $T=2.08$ K with low excitation. The lower (black) dashed line corresponds to $Q\propto f^{3/2}$, and the different symbols represent the results of different experiments. The upper (red) dotted line indicates the calculated resonator $Q$ if it were governed solely by bulk dissipation of the second-sound waves.

Figure 2

(a) Resonance curves obtained by slowly sweeping the frequency detuning $f_G$ of the harmonic driving force near the 7th resonance at $T=2.08\text{K}$. The stationary standing-wave response amplitude $U_B$ is plotted for two different fixed driving amplitudes: $U_G=4$ V shown by the (red) triangles, and $U_G=2$ V shown by the (blue) circles. The (blue) line represents a rescaling of the latter response to facilitate comparison of the widths of the two resonances. The horizontal dashed (green and blue) lines indicate levels of $1/\sqrt{2}$ of the peak value, to provide indicators of the quality $Q$ factor. (b) Time dependence of the amplitudes of the second-sound signals after cessation of external excitation $U_G$ of resonator. The (black) lines represent least-squares fits to the data in each case.

Figure 3

(a) Time dependence of the amplitudes of the harmonics during the decay of the signal, following switching off of the drive at $t=0$ under stationary conditions: 11th resonance, $f_G=516.81$ Hz, $T=2.08$ K, and $U=3$ V corresponding to $W=10.8$ mW/cm${}^2$. The (red) dashed line represents a fit to the linear part of the decaying main harmonic (at the resonant frequency), and the vertical (red) arrow and associated dotted lines indicate the end point of the nonlinear decay. (b) An expansion of the data to show in more detail the chaotic behavior of the higher harmonics just after switching off the driving force. The double-headed arrows indicate correlated maxima/minima at the resonant frequency/second harmonic.

Figure 4

The energy of nonlinear decay, defined as the difference between the energy of the main harmonic and that in the linear (exponential) decay.

Figure 5

Free decay of the developed inverse cascade following excitation on the 96th resonance at 4715 Hz with $U_G=5$ V, $q=30$ mW cm${}^{-2}$, and $T=2.08$ K. (a) Full-length time domain signal $U_B\left(t\right)$. (b)–(d) Short clips from the time domain signal, taken at the positions indicated by the arrows in (a) from left to right, respectively. (e) Evolution with time of the energy in the different left cascade (LC) subharmonics, higher harmonics, and combinational frequencies, obtained from a windowed Fourier analysis of $U_B\left(t\right)$: main harmonic (red) crosses; second harmonic (red) squares. Full lines indicate the behavior of the subharmonics, corresponding to the $\sim$29th (LC1), $\sim$38th (LC2), $\sim$59th (LC3), $\sim$77th (LC4), and $\sim$87th (LC5) resonances. The combinational frequencies corresponding to interaction between main and LC harmonics are shown as thin dashed lines).